https://neckpinch.com/authors/liam-mazurowski/New arXiv papers by Liam Mazurowski, from Neckpinch Tue, 03 Nov 2026 08:37:45 GMT https://neckpinch.com/papers/2502.18455/arxiv:2502.18455Tue, 25 Feb 2025 00:00:00 GMTLiam Mazurowski, Jintian Zhu — We prove the existence of compact surfaces with prescribed constant mean curvature in asymptotically flat and asymptotically hyperbolic manifolds. More precisely, let (M³,g) be an asymptotically flat manifold with scalar curvature R≥ 0. Then, for each constant c>0, there exists a compact, almost-embedded, free boundary constant mean curvature surface Σ⊂ M with mean curvature c. Likewise, let (M³,g) be an asymptotically hyperbolic manifold with scalar curvature R≥ −6. Then, for each constant c>2, there exists a compact, almost-embedded, free boundary constant mean curvature surface Σ⊂ M with mean curvature c. The proof combines min-max theory with the following fact about inverse mean curvature flow which is of independent interest: for any T the inverse mean curvature flow emerging out of a point p far enough out in an asymptotically flat (or asymptotically hyperbolic) end will remain smooth for all times t∈ (-∞,T].Mean curvature flowOther curvature flowsInverse mean curvature flowScalar curvatureBoundaries & non-compactFree boundary & capillaryMinimal & CMC surfaces https://neckpinch.com/papers/2305.19784/arxiv:2305.19784Wed, 31 May 2023 00:00:00 GMTLiam Mazurowski, Xuan Yao — Consider a complete asymptotically flat 3-manifold M with non-negative scalar curvature and non-empty minimal boundary Σ. Fix a number 1 < p < 3. We derive monotone quantities for p-harmonic functions on M which become constant on Schwarzschild. These monotonicity formulas imply a sharp mass-capacity estimate relating the ADM mass of M with the p-capacity of Σ in M, which was first proved by Xiao using weak inverse mean curvature flow.Mean curvature flowOther curvature flowsInverse mean curvature flow3-manifoldsScalar curvatureBoundaries & non-compactGeometric inequalitiesPhysics